How to use an abacus with Math Mammoth

Recently I've received several questions about abacus usage within Math Mammoth curriculum. Here is what I wrote and added to my FAQ on the site.

The only way the abacus is used in my books is where each bead counts as one. Nothing fancy. It is NOT used like Chinese, Russian, or any of the other abaci where one bead might count as 5, 10, or 100.

A 100-bead abacus or school abacus simply contains 10 beads on 10 rods, a total of 100. In the school abacus, each bead simply represents one. The 100-bead abacus lets children both "see" the numbers and use their touch while making them.

First and foremost, the abacus is used in the place value section in 1st grade where children learn about tens and ones (numbers up to 100). We use it to show clearly how 45 is made up of 4 tens and 5 ones, for example.

Secondly, you can use the abacus with addition and subtraction problems in 1st and 2nd grades. For example:

• Show the child additions and subtractions with whole tens. For example, to solve 50 + 20, first make 50 on the abacus. Then add 20 more.
• Add a two-digit number and a single-digit number. For example, to solve 23 + 5, first make 23 on the abacus. Then add five beads.
• Show some "shortcuts" in addition or subtraction. For example, to solve 34 + 20, first make 34 on the abacus. To add 20, add two whole rows of beads. Then the student checks how many whole tens and how many individual beads is the total.
  
  Or, to solve 85 − 20, first make 85. Then pull back two whole rows of beads.
  
  Or, to add 23 + 44. First make 23. Then make 44 on using the five lowest rows of the abacus. Have the child now count the whole tens (6), and the individual beads from the two rows (3 + 4). This shows adding the tens separately, and adding the ones separately. From this you can graduate to making first 23, then adding 4 full rows of beads for 40, and then adding 4 individual beads from the same row as the 3 beads.

The purpose is mainly to help children to visualize two-digit numbers, and to add and subtract two-digit numbers.

The goal in my books is to drop the abacus by 3rd grade. Even before that, students use visual models, and from those go on to the abstract. The quicker the child can use visual models, and then do the math problems without any models, the better.

At Amazon you can find Melissa & Doug Classic Wooden Abacus for around $12. An abacus where the beads alternate colors by fives is even more useful (but may be out of stock).

Browse Amazon's abacus selection here. Other stores carry abaci as well.

You can also use this virtual abacus. Or, make your own abacus. Just don't make it exactly like they show on that web page but instead use 10 bamboo skewer with 10 beads in each so you get a 10 x 10 abacus.

How to use an abacus with Math Mammoth

Recently I've received several questions about abacus usage within Math Mammoth curriculum. Here is what I wrote and added to my FAQ on the site.

The only way the abacus is used in my books is where each bead counts as one. Nothing fancy. It is NOT used like Chinese, Russian, or any of the other abaci where one bead might count as 5, 10, or 100.

A 100-bead abacus or school abacus simply contains 10 beads on 10 rods, a total of 100. In the school abacus, each bead simply represents one. The 100-bead abacus lets children both "see" the numbers and use their touch while making them.

First and foremost, the abacus is used in the place value section in 1st grade where children learn about tens and ones (numbers up to 100). We use it to show clearly how 45 is made up of 4 tens and 5 ones, for example.

Secondly, you can use the abacus with addition and subtraction problems in 1st and 2nd grades. For example:

• Show the child additions and subtractions with whole tens. For example, to solve 50 + 20, first make 50 on the abacus. Then add 20 more.
• Add a two-digit number and a single-digit number. For example, to solve 23 + 5, first make 23 on the abacus. Then add five beads.
• Show some "shortcuts" in addition or subtraction. For example, to solve 34 + 20, first make 34 on the abacus. To add 20, add two whole rows of beads. Then the student checks how many whole tens and how many individual beads is the total.
  
  Or, to solve 85 − 20, first make 85. Then pull back two whole rows of beads.
  
  Or, to add 23 + 44. First make 23. Then make 44 on using the five lowest rows of the abacus. Have the child now count the whole tens (6), and the individual beads from the two rows (3 + 4). This shows adding the tens separately, and adding the ones separately. From this you can graduate to making first 23, then adding 4 full rows of beads for 40, and then adding 4 individual beads from the same row as the 3 beads.

The purpose is mainly to help children to visualize two-digit numbers, and to add and subtract two-digit numbers.

The goal in my books is to drop the abacus by 3rd grade. Even before that, students use visual models, and from those go on to the abstract. The quicker the child can use visual models, and then do the math problems without any models, the better.

At Amazon you can find Melissa & Doug Classic Wooden Abacus for around $12. An abacus where the beads alternate colors by fives is even more useful (but may be out of stock).

Browse Amazon's abacus selection here. Other stores carry abaci as well.

You can also use this virtual abacus. Or, make your own abacus. Just don't make it exactly like they show on that web page but instead use 10 bamboo skewer with 10 beads in each so you get a 10 x 10 abacus.

Abacus and basic division concept

Let's be reminded first of all that there are two basic "interpretations" of division:

1) Sharing division. In it, you think that 16 / 2 means "If there are 16 beads divided between 2 people, how many does each one get?

2) Quotative division. In it, you think how many groups of the same size you can form. Or, "how many times does the divisor fit into the dividend?" For example, 16 / 2 is interpreted as: "If you have 16 beads, how many groups of 2 beads can you make?"

Sharing division is easy to understand as a concept, but it's hard to do without the knowledge of multiplication tables. Just imagine, if a child who doesn't know their times tables is trying to find the answer to the question, "If there are 56 strawberries and 8 people, how many will each one get?" It'll take some guesswork, trying and checking.

But quotative division is easy for children to do with the help of manipulatives (or by drawing pictures). Just make a group of 56, and start forming groups of 8 out of it. Once you're done, check how many groups you got. There's no guessing. It's just repeated subtraction - each group the child forms is "subtracted" or set apart.

For this reason I have decided to start off with quotative division in my Division 1 book. Children can first just partition off groups of certain size from the picture problems and count how many groups they got.

And this activitity lends itself perfectly to the 100-bead abacus, as well.

For example, we have 20 / 4.

Take 20 beads:

| 0 0 0 0 0 0 0 0 0 0    |
| 0 0 0 0 0 0 0 0 0 0    |

Group them in groups of 4.

| 0000  0000   00    |
| 0000  0000   00    |

Note the one group is formed from 2 beads on one bar and 2 on the other. Answer: 5 groups.

Another example: 27 / 9

First take 27 beads.

| 0 0 0 0 0 0 0 0 0 0                |
| 0 0 0 0 0 0 0 0 0 0                |
| 0 0 0 0 0 0 0               0 0 0  |

Form one group of 9 from the first bar (green), another from the second bar (blue). The third group is the purple ones.

| 0     0 0 0 0 0 0 0 0 0            |
| 0     0 0 0 0 0 0 0 0 0            |
| 0 0 0 0 0 0 0               0 0 0  |

Groups of fives are particularly interesting since they look so "orderly" on the abacus.

Ask your student, like I did today, what's 90 / 5?

Well, you organize your 90 beads like this, and answer is easily counted as 18 groups.

| 0 0 0 0 0        0 0 0 0 0  |
| 0 0 0 0 0        0 0 0 0 0  |
| 0 0 0 0 0        0 0 0 0 0  |
| 0 0 0 0 0        0 0 0 0 0  |
| 0 0 0 0 0        0 0 0 0 0  |
| 0 0 0 0 0        0 0 0 0 0  |
| 0 0 0 0 0        0 0 0 0 0  |
| 0 0 0 0 0        0 0 0 0 0  |
| 0 0 0 0 0        0 0 0 0 0  |

You can, of course, use abacus for sharing division, too. But make easy problems where the sharing is easy to do, such as 20 / 2 or 24 / 2 or 15 / 3.

Once the concepts of quotative division and sharing division are clear, you need to spend a lot of time with multiplication/division connection, because that is what children will use to solve most division problems - they'll need to get away from using manipulatives after the initial stages.

Abacus and basic division concept

Let's be reminded first of all that there are two basic "interpretations" of division:

1) Sharing division. In it, you think that 16 / 2 means "If there are 16 beads divided between 2 people, how many does each one get?

2) Quotative division. In it, you think how many groups of the same size you can form. Or, "how many times does the divisor fit into the dividend?" For example, 16 / 2 is interpreted as: "If you have 16 beads, how many groups of 2 beads can you make?"

Sharing division is easy to understand as a concept, but it's hard to do without the knowledge of multiplication tables. Just imagine, if a child who doesn't know their times tables is trying to find the answer to the question, "If there are 56 strawberries and 8 people, how many will each one get?" It'll take some guesswork, trying and checking.

But quotative division is easy for children to do with the help of manipulatives (or by drawing pictures). Just make a group of 56, and start forming groups of 8 out of it. Once you're done, check how many groups you got. There's no guessing. It's just repeated subtraction - each group the child forms is "subtracted" or set apart.

For this reason I have decided to start off with quotative division in my Division 1 book. Children can first just partition off groups of certain size from the picture problems and count how many groups they got.

And this activitity lends itself perfectly to the 100-bead abacus, as well.

For example, we have 20 / 4.

Take 20 beads:

| 0 0 0 0 0 0 0 0 0 0    |
| 0 0 0 0 0 0 0 0 0 0    |

Group them in groups of 4.

| 0000  0000   00    |
| 0000  0000   00    |

Note the one group is formed from 2 beads on one bar and 2 on the other. Answer: 5 groups.

Another example: 27 / 9

First take 27 beads.

| 0 0 0 0 0 0 0 0 0 0                |
| 0 0 0 0 0 0 0 0 0 0                |
| 0 0 0 0 0 0 0               0 0 0  |

Form one group of 9 from the first bar (green), another from the second bar (blue). The third group is the purple ones.

| 0     0 0 0 0 0 0 0 0 0            |
| 0     0 0 0 0 0 0 0 0 0            |
| 0 0 0 0 0 0 0               0 0 0  |

Groups of fives are particularly interesting since they look so "orderly" on the abacus.

Ask your student, like I did today, what's 90 / 5?

Well, you organize your 90 beads like this, and answer is easily counted as 18 groups.

| 0 0 0 0 0        0 0 0 0 0  |
| 0 0 0 0 0        0 0 0 0 0  |
| 0 0 0 0 0        0 0 0 0 0  |
| 0 0 0 0 0        0 0 0 0 0  |
| 0 0 0 0 0        0 0 0 0 0  |
| 0 0 0 0 0        0 0 0 0 0  |
| 0 0 0 0 0        0 0 0 0 0  |
| 0 0 0 0 0        0 0 0 0 0  |
| 0 0 0 0 0        0 0 0 0 0  |

You can, of course, use abacus for sharing division, too. But make easy problems where the sharing is easy to do, such as 20 / 2 or 24 / 2 or 15 / 3.

Once the concepts of quotative division and sharing division are clear, you need to spend a lot of time with multiplication/division connection, because that is what children will use to solve most division problems - they'll need to get away from using manipulatives after the initial stages.

An AHA! abacus moment in the life of a preschooler

I've been doing math lessons with my 3-year old using the 100-bead abacus. Usually we do a few problems where she tells me a number to make and I tell her a number to make, back and forth. Today she asked me to make 51, and I asked her to make 68. These went smoothly since she's getting pretty good at this now.

Then we did a few "more than" problems. I said, "Let's say your sister has 5 cookies and you have one more than her. How many do you have?" This is a new concept to her so we need to do it slowly and carefully with the abacus: first make her sister's cookies, then let her have the same amount, then give her one more.

Then we do a few subtraction problems such as 7 − 4. She moved 7 beads, then "took away" or moved the other way 4 beads, and how many were left? 3 beads. I showed her also 50 − 10 = 40.

She started doing her own problem, "Let's do 9 − ..." and while she was thinking, I quickly proposed "... minus nine". Nine minus nine. Well, she moved nine beads, then COUNTED the beads one by one moving them the other direction, and was left with none... and what a SURPRISE it was to her! She had to start giggling!

I immediately showed her another one, 4 − 4. She did 10 − 10 herself. And THEN I showed her 100 − 100, which made the greatest giggles of all!

It was just so cute so I had to share. Plus, now you know several ways how to teach math concepts with the abacus.

An AHA! abacus moment in the life of a preschooler

I've been doing math lessons with my 3-year old using the 100-bead abacus. Usually we do a few problems where she tells me a number to make and I tell her a number to make, back and forth. Today she asked me to make 51, and I asked her to make 68. These went smoothly since she's getting pretty good at this now.

Then we did a few "more than" problems. I said, "Let's say your sister has 5 cookies and you have one more than her. How many do you have?" This is a new concept to her so we need to do it slowly and carefully with the abacus: first make her sister's cookies, then let her have the same amount, then give her one more.

Then we do a few subtraction problems such as 7 − 4. She moved 7 beads, then "took away" or moved the other way 4 beads, and how many were left? 3 beads. I showed her also 50 − 10 = 40.

She started doing her own problem, "Let's do 9 − ..." and while she was thinking, I quickly proposed "... minus nine". Nine minus nine. Well, she moved nine beads, then COUNTED the beads one by one moving them the other direction, and was left with none... and what a SURPRISE it was to her! She had to start giggling!

I immediately showed her another one, 4 − 4. She did 10 − 10 herself. And THEN I showed her 100 − 100, which made the greatest giggles of all!

It was just so cute so I had to share. Plus, now you know several ways how to teach math concepts with the abacus.

Basic abacus as a manipulative

I didn't want to leave out one of the best manipulatives there is for grades 1-2: just a simple "school" abacus that has 10 wires and 10 beads on each wire.

I'm not talking about a Chinese or Japanese abacus with a special counting system.

I'm talking about just using this simple abacus for counting, and treating each bead as 1. You don't have to learn any of these sophisticated systems that have been in use with various abacuses. Just consider each bead being 1, period. Then you have essentially 10 tens, or a hundred, in your abacus.

And that goes a long way explaining tens and ones or 2-digit place value on 1st grade.

You can also show the child things such as similarities in
10 − 5
20 − 5
60 − 5

or let the student find sums of 2-digit numbers: 23 + 45. He can move 2 tens and 4 tens, then 3 and 5 individual pieces - so the abacus can model adding the tens and ones separately.

You can let the child explore what happens with 28 + 9.

It's better if the abacus has the first five beads colored differently from the next five, in each row, like in this silly picture.

Then the child will easily recognize 6, 7, and 8 beads without counting. Also, let's say you'd choose 6 beads on one wire and 8 on the next one. You can show how the five and five on those two wires makes ten, and some are left over.

You can also model multiplication: move for example 4 beads on each of the 5 neighboring wires and you have 5 times 4.

So this is not rocket science; it is very easy. No need to learn any new systems.

Here's a picture of an old school abacus.

Wikipedia has info on all different kinds of abaci, including this kind of usage of the school abacus.

You can browse Amazon's abacus selection here.

Basic abacus as a manipulative

I didn't want to leave out one of the best manipulatives there is for grades 1-2: just a simple "school" abacus that has 10 wires and 10 beads on each wire.

I'm not talking about a Chinese or Japanese abacus with a special counting system.

I'm talking about just using this simple abacus for counting, and treating each bead as 1. You don't have to learn any of these sophisticated systems that have been in use with various abacuses. Just consider each bead being 1, period. Then you have essentially 10 tens, or a hundred, in your abacus.

And that goes a long way explaining tens and ones or 2-digit place value on 1st grade.

You can also show the child things such as similarities in
10 − 5
20 − 5
60 − 5

or let the student find sums of 2-digit numbers: 23 + 45. He can move 2 tens and 4 tens, then 3 and 5 individual pieces - so the abacus can model adding the tens and ones separately.

You can let the child explore what happens with 28 + 9.

It's better if the abacus has the first five beads colored differently from the next five, in each row, like in this silly picture.

Then the child will easily recognize 6, 7, and 8 beads without counting. Also, let's say you'd choose 6 beads on one wire and 8 on the next one. You can show how the five and five on those two wires makes ten, and some are left over.

You can also model multiplication: move for example 4 beads on each of the 5 neighboring wires and you have 5 times 4.

So this is not rocket science; it is very easy. No need to learn any new systems.

Here's a picture of an old school abacus.

Wikipedia has info on all different kinds of abaci, including this kind of usage of the school abacus.

You can browse Amazon's abacus selection here.